High-Fidelity Universal Gate Set for $^9$Be$^+$ Ion Qubits

We report high-fidelity laser-beam-induced quantum logic gates on
magnetic-field-insensitive qubits comprised of hyperfine states in $^{9}$Be$^+$
ions with a memory coherence time of more than 1 s. We demonstrate single-qubit
gates with error per gate of $3.8(1)\times 10^{-5}$. By creating a Bell state
with a deterministic two-qubit gate, we deduce a gate error of
$8(4)\times10^{-4}$. We characterize the errors in our implementation and
discuss methods to further reduce imperfections towards values that are
compatible with fault-tolerant processing at realistic overhead.